## EPR Challenge

Related to physics

### EPR Challenge

Here is a challenge that might interest the calculus experts out there. Here are the formulas to be considered from a Mathematica notebook file.

Code: Select all
(* Michel Fodje's Minkwe simulationtranslated from Python to Mathematica by John Reed13 Nov 2013 *)(* Set run time parameters, initialize arrays *)trials=5000000;aliceDeg=ConstantArray[0,trials];bobDeg=ConstantArray[0,trials];aliceDet=ConstantArray[0,trials];bobDet=ConstantArray[0,trials];nPP=ConstantArray[0,361];nNN=ConstantArray[0,361];nPN=ConstantArray[0,361];nNP=ConstantArray[0,361];nA=ConstantArray[0,361];nB=ConstantArray[0,361];(* Detector test function *)test[angle_,e_,\[Lambda]_]:=Module[{c,out},c=-Cos[1(angle-e)];If[\[Lambda]>=Abs[c],out=0,out=Sign[c]];out](* Generate particle data  *)Do[eVector=RandomReal[{0,2 \[Pi]}];\[Lambda]=1/2 Sin[RandomReal[{0,\[Pi](0.5)}]]^2;eLeft=RandomReal[{0,2 \[Pi]}];eRight=eLeft+\[Pi];aliceAngle=RandomReal[{0,2 \[Pi]}];aliceDeg[[i]]=aliceAngle/Degree;bobAngle=RandomReal[{0,2 \[Pi]}];bobDeg[[i]]=bobAngle/Degree;aliceDet[[i]]=test[aliceAngle,eLeft,\[Lambda]];bobDet[[i]]=test[bobAngle,eRight,\[Lambda]],{i,trials}](* statistical analysis of particle data  *)Do[\[Theta]=Ceiling[(aliceDeg[[i]]-bobDeg[[i]])]-1;aliceD=aliceDet[[i]];bobD=bobDet[[i]]; If[aliceD==1,nA[[\[Theta]]]++];If[bobD==1,nB[[\[Theta]]]++];If[aliceD==1&&bobD==1,nPP[[\[Theta]]]++];If[aliceD==1&&bobD==-1,nPN[[\[Theta]]]++];If[aliceD==-1&&bobD==1,nNP[[\[Theta]]]++];If[aliceD==-1&&bobD==-1,nNN[[\[Theta]]]++],{i,trials}](* Calculate mean values and plot *)pPP=0; pPN=0; pNP=0;pNN=0;mean=ConstantArray[0,361];Do[sum=nPP[[i]]+nPN[[i]]+nNP[[i]]+nNN[[i]];If[sum==0,Goto[jump],{pPP=nPP[[i]]/sum;       pNP=nNP[[i]]/sum;       pPN=nPN[[i]]/sum;       pNN=nNN[[i]]/sum;mean[[i]]=pPP+pNN-pPN-pNP}];Label[jump],{i,361}]simulation=ListPlot[mean]cos=Plot[-Cos[x Degree],{x,0,360},PlotStyle->{Red,Thick}];(* Compare mean values with Cosine *)Show[simulation,cos]

Prove or not prove that if the number of trials goes to infinity and the degree increment goes to zero that we get the result of -a.b where a is the vector represented by aliceAngle above and b is represented by bobAngle above. Basically that we get a pure -cos curve.
FrediFizzx
Independent Physics Researcher

Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA

### Re: EPR Challenge

Let's see how this works here.

$\lim_{n\to \infty } \, \frac{\sum _{i=1}^n A^i B^i}{n}$

So that is the basic function we need to calculate with A and B defined as Joy has shown here.
...
FrediFizzx
Independent Physics Researcher

Posts: 2905
Joined: Tue Mar 19, 2013 7:12 pm
Location: N. California, USA